This happened to me whilst working on a pre-existing site recently.
The Problem
You are trying to install WordPress onto a subdirectory of an existing site that uses the .htaccess files for routing. When you try to add in the changes needed for WordPress to work you break the existing site.
The Cause
WordPress .htaccess files do not play nicely with other .htaccess files
The Solution
Simply add in the WordPress generated .htaccess files in the WordPress directory, and everything magically works.
For Unix people this must be obvious, but for me it was surprisingly brilliant.